Wave turbulence and intermittency

In the early 1960s, it was established that the stochastic initial value pr oblem for weakly coupled wave systems has a natural asymptotic closure indu ced by the dispersive properties of the waves and the large separation of l inear and nonlinear time scales. One is thereby led to kinetic equations fo r the redistribution of spectral densities via three- and four-wave resonan ces together with a nonlinear renormalization of the frequency. The kinetic equations have equilibrium solutions which are much richer than the famili ar thermodynamic, Fermi-Dirac or Bose-Einstein spectra and admit in additio n finite flux (Kolmogorov-Zakharov) solutions which describe the transfer o f conserved densities (e.g. energy) between sources and sinks. There is muc h one can learn from the kinetic equations about the behavior of particular systems of interest including insights in connection with the phenomenon o f intermittency. What we would like to convince you is that what we call we ak or wave turbulence is every bit as rich as the macho turbulence of 3D hy drodynamics at high Reynolds numbers and, moreover, is analytically more tr actable. It is an excellent paradigm for the study of many-body Hamiltonian systems which are driven far from equilibrium by the presence of external forcing and damping. In almost all cases, it contains within its solutions behavior which invalidates the premises on which the theory is based in som e spectral range. We give some new results concerning the dynamic breakdown of the weak turbulence description and discuss the fully nonlinear and int ermittent behavior which follows. These results may also be important for p roving or disproving the global existence of solutions for the underlying p artial differential equations. Wave turbulence is a subject to which many h ave made important contributions. But no contributions have been more funda mental than those of Volodja Zakharov whose 60th birthday we celebrate at t his meeting. He was the first to appreciate that the kinetic equations admi t a far richer class of solutions than the fluxless thermodynamic solutions of equilibrium systems and to realize the central roles that finite flux s olutions play in non-equilibrium systems. It is appropriate, therefore, tha t we call these Kolmogorov-Zakharov (KZ) spectra. (C) 2001 Elsevier Science B.V. All rights reserved.